Properties

Label 216.2.d
Level $216$
Weight $2$
Character orbit 216.d
Rep. character $\chi_{216}(109,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $3$
Sturm bound $72$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 216.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(216, [\chi])\).

Total New Old
Modular forms 42 16 26
Cusp forms 30 16 14
Eisenstein series 12 0 12

Trace form

\( 16q + 2q^{4} + O(q^{10}) \) \( 16q + 2q^{4} + 2q^{10} + 18q^{16} - 26q^{22} - 16q^{25} - 14q^{28} + 8q^{31} - 20q^{34} - 30q^{40} + 4q^{46} + 24q^{49} - 8q^{52} - 16q^{55} + 28q^{58} - 10q^{64} + 66q^{70} - 8q^{73} + 80q^{76} - 72q^{79} - 52q^{82} + 70q^{88} - 48q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(216, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
216.2.d.a \(4\) \(1.725\) \(\Q(\zeta_{8})\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(-4\) \(q+\zeta_{8}^{2}q^{2}-2q^{4}+(\zeta_{8}-\zeta_{8}^{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
216.2.d.b \(4\) \(1.725\) \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(0\) \(4\) \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+q^{7}+\cdots\)
216.2.d.c \(8\) \(1.725\) 8.0.629407744.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{4}q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(216, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(216, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T^{2} )^{2} \))(\( 1 - 3 T^{2} + 4 T^{4} \))(\( 1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8} \))
$3$ 1
$5$ (\( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \))(\( ( 1 - 9 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 8 T^{2} + 38 T^{4} - 200 T^{6} + 625 T^{8} )^{2} \))
$7$ (\( ( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - T + 7 T^{2} )^{4} \))(\( ( 1 + 7 T^{2} + 49 T^{4} )^{4} \))
$11$ (\( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \))(\( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 24 T^{2} + 358 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2} \))
$13$ (\( ( 1 - 13 T^{2} )^{4} \))(\( ( 1 + 2 T^{2} + 169 T^{4} )^{2} \))(\( ( 1 - 30 T^{2} + 451 T^{4} - 5070 T^{6} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 + 17 T^{2} )^{4} \))(\( ( 1 + 6 T^{2} + 289 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} + 614 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 19 T^{2} )^{4} \))(\( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \))(\( ( 1 - 18 T^{2} + 691 T^{4} - 6498 T^{6} + 130321 T^{8} )^{2} \))
$23$ (\( ( 1 + 23 T^{2} )^{4} \))(\( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 16 T^{2} - 250 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \))(\( ( 1 - 36 T^{2} + 1558 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2} \))(\( ( 1 + 7 T + 31 T^{2} )^{4} \))(\( ( 1 - 2 T + 31 T^{2} )^{8} \))
$37$ (\( ( 1 - 37 T^{2} )^{4} \))(\( ( 1 - 46 T^{2} + 1369 T^{4} )^{2} \))(\( ( 1 - 102 T^{2} + 4891 T^{4} - 139638 T^{6} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 + 41 T^{2} )^{4} \))(\( ( 1 + 54 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 + 4 T^{2} + 1574 T^{4} + 6724 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 - 43 T^{2} )^{4} \))(\( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 - 108 T^{2} + 6166 T^{4} - 199692 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 47 T^{2} )^{4} \))(\( ( 1 + 80 T^{2} + 3750 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} \))(\( ( 1 - 25 T^{2} + 2809 T^{4} )^{2} \))(\( ( 1 - 132 T^{2} + 9526 T^{4} - 370788 T^{6} + 7890481 T^{8} )^{2} \))
$59$ (\( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 102 T^{2} + 3481 T^{4} )^{2} \))(\( ( 1 - 56 T^{2} + 6374 T^{4} - 194936 T^{6} + 12117361 T^{8} )^{2} \))
$61$ (\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{4} \))(\( ( 1 - 46 T^{2} - 1101 T^{4} - 171166 T^{6} + 13845841 T^{8} )^{2} \))
$67$ (\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 67 T^{2} )^{4} \))(\( ( 1 - 125 T^{2} + 4489 T^{4} )^{4} \))
$71$ (\( ( 1 + 71 T^{2} )^{4} \))(\( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 71 T^{2} )^{8} \))
$73$ (\( ( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{2} \))(\( ( 1 - 3 T + 73 T^{2} )^{4} \))(\( ( 1 - 2 T + 35 T^{2} - 146 T^{3} + 5329 T^{4} )^{4} \))
$79$ (\( ( 1 + 10 T + 79 T^{2} )^{4} \))(\( ( 1 - 4 T + 79 T^{2} )^{4} \))(\( ( 1 + 12 T + 187 T^{2} + 948 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( 1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8} \))(\( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \))(\( ( 1 - 236 T^{2} + 25910 T^{4} - 1625804 T^{6} + 47458321 T^{8} )^{2} \))
$89$ (\( ( 1 + 89 T^{2} )^{4} \))(\( ( 1 + 66 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 304 T^{2} + 38918 T^{4} + 2407984 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 7 T + 97 T^{2} )^{4} \))(\( ( 1 + 6 T + 175 T^{2} + 582 T^{3} + 9409 T^{4} )^{4} \))
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